Z-Score Formula and Worked Examples

The formula

A z-score tells you how many standard deviations a value sits from the mean. The formula is z = (x − μ) / σ, where x is the value, μ is the mean and σ is the standard deviation. When you work from a sample, the sample mean x̄ and sample standard deviation s take their place.

Standardizing in this way turns any normal distribution into the standard normal distribution, which has a mean of 0 and a standard deviation of 1. That is why one table serves every normal problem.

Reading the sign and size

Values above the mean have positive z-scores and values below the mean have negative z-scores. A z-score of 0 is exactly at the mean.

• z = 1.00 means one standard deviation above the mean.
• z = −1.23 means 1.23 standard deviations below the mean.
• The larger the absolute value of z, the less common the value is under a normal model.

Example 1: a value above the mean

A test has a mean of 100 and a standard deviation of 15. A score of 115 gives z = (115 − 100) / 15 = 1.00.

To find the percentile, look up Φ(1.00) in a left-tail table: 0.8413. The score is at about the 84th percentile, meaning roughly 84% of scores are at or below it.

Example 2: a value below the mean

A different test has a mean of 500 and a standard deviation of 100. A score of 377 gives z = (377 − 500) / 100 = −1.23.

Φ(−1.23) = 0.1093, so about 10.9% of scores fall at or below 377. The share of scores above 377 is 1 − 0.1093 = 0.8907.

The z-score for a sample mean

When the value you are standardizing is a sample mean, the spread you divide by is the standard error, σ / √n. The formula becomes z = (x̄ − μ) / (σ / √n).

Example: the population mean is 100 and σ = 15. A sample of n = 36 has x̄ = 105. The standard error is 15 / √36 = 2.5, so z = (105 − 100) / 2.5 = 2.00. Sample means vary less than single values, which is why the denominator shrinks as n grows.

When the normal model is reasonable

A z-score always can be calculated, but a percentile from the normal table is only trustworthy when the data are roughly bell-shaped. Check a histogram first. For sample means, the distribution is approximately normal when the sample size is large enough, even if the population is not normal. Heavily skewed data with a small sample are a poor fit for this method.